Form Of Conservation Laws Research Articles

A kinetic energy preserving DG scheme applied to mechanical FSI

AbstractIn the context of mechanical fluid‐structure interaction (FSI) comprising moving or deforming structures, fluid discretizations need to cope with time‐dependent fluid domains and resulting grid deformations in addition to the general challenges regarding e.g. boundary layers and turbulent phenomena. Recent approaches in the simulation of compressible turbulent flow are based on so‐called split forms of conservation laws to guarantee the preservation of secondary physical quantities such as kinetic energy. For the simulation of turbulent flows, this often leads to a better representation of the kinetic energy spectrum. Initially, kinetic energy preserving(KEP) DG schemes have been constructed on Gauss‐Legendre‐Lobatto(GLL) nodes containing the interval end points, however, KEP DG schemes based on the classical Gauss‐Legendre(GL) nodes are potentially more accurate and may be also more efficient than its GLL variant for certain applications. In this work, the KEP‐DG schemes both on GL and GLL nodes are applied to the classical moving piston test case via an ALE formulation on moving fluid grids showing a more accurate frequency representation of the structure displacement in case of GLL nodes. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

GPU driven finite difference WENO scheme for real time solution of the shallow water equations

The shallow water equations are applicable to many common engineering problems involving modelling of waves dominated by motions in the horizontal directions (e.g. tsunami propagation, dam breaks). As such events pose substantial economic costs, as well as potential loss of life, accurate real-time simulation and visualization methods are of great importance. For this purpose, we propose a new finite difference scheme for the 2D shallow water equations that is specifically formulated to take advantage of modern GPUs. The new scheme is based on the so-called Picard integral formulation of conservation laws combined with Weighted Essentially Non-Oscillatory reconstruction. The emphasis of the work is on third order in space and second order in time solutions (in both single and double precision). Further, the scheme is well-balanced for bathymetry functions that are not surface piercing and can handle wetting and drying in a GPU-friendly manner without resorting to long and specific case-by-case procedures. We also present a fast single kernel GPU implementation with a novel boundary condition application technique that allows for simultaneous real-time visualization and single precision simulations even on large ( > 2000 × 2000) grids on consumer-level hardware - the full kernel source codes are also provided online at https://github.com/pparna/swe_pifweno3.

Nonlocal weakly relativistic permittivity tensor of magnetized plasma near electron cyclotron resonances

Compact expressions are derived for the nonlocal permittivity tensor of weakly relativistic plasma in a 2D nonuniform magnetic field near the resonances at the second harmonic of the electron cyclotron frequency for an extraordinary wave and at the first harmonic for an ordinary wave. It is shown that the wave equation with allowance for the obtained thermal correction to the permittivity tensor in the form of a differential operator in transverse (with respect to the external magnetic field) coordinates possesses an integral in the form of the energy conservation law.

Axial magnetic field and toroidally streaming fast ions in the dense plasma focus are natural consequences of conservation laws in the curved axisymmetric geometry of the current sheath. II. Towards a first principles theory

This paper continues earlier discussion [S. K. H. Auluck, Phys. Plasmas 21, 102515 (2014)] concerning the formulation of conservation laws of mass, momentum, and energy in a local curvilinear coordinate system in the dense plasma focus. This formulation makes use of the revised Gratton-Vargas snowplow model [S. K. H. Auluck, Phys. Plasmas 20, 112501 (2013)], which provides an analytically defined imaginary surface in three dimensions which resembles the experimentally determined shape of the plasma. Unit vectors along the local tangent to this surface, along the azimuth, and along the local normal define a right-handed orthogonal local curvilinear coordinate system. The simplifying assumption that physical quantities have significant variation only along the normal enables writing laws of conservation of mass, momentum, and energy in the form of effectively one-dimensional hyperbolic conservation law equations using expressions for various differential operators derived for this coordinate system. This formulation demonstrates the highly non-trivial result that the axial magnetic field and toroidally streaming fast ions, experimentally observed by multiple prestigious laboratories, are natural consequences of conservation of mass, momentum, and energy in the curved geometry of the dense plasma focus current sheath. The present paper continues the discussion in the context of a 3-region shock structure similar to the one experimentally observed: an unperturbed region followed by a hydrodynamic shock containing some current followed by a magnetic piston. Rankine-Hugoniot conditions are derived, and expressions are obtained for the specific volumes and pressures using the mass-flux between the hydrodynamic shock and the magnetic piston and current fraction in the hydrodynamic shock as unknown parameters. For the special case of a magnetic piston that remains continuously in contact with the fluid being pushed, the theory gives closed form algebraic results for the fraction of current flowing in the hydrodynamic shock, specific volume, pressure, and fluid velocity of the hydrodynamic shock region, the tangential, normal, and azimuthal components of velocity in the magnetized plasma, the density of the magnetized plasma, the normal and tangential components of the magnetic field, and the tangential, normal, and azimuthal components of the electric field. This explains the occurrence of azimuthally streaming high energy deuterons experimentally observed by Frascati and Stuttgart. The expression derived for the azimuthal component of vector potential can serve as the basis for a proposed experimental test of the theory.

On classical integrability of the hydrodynamics of quantum integrable systems

Recently, a hydrodynamic description of local equilibrium dynamics in quantum integrable systems was discovered. In the diffusionless limit, this is equivalent to a certain ‘Bethe–Boltzmann’ kinetic equation, which has the form of an integro-differential conservation law in D. The purpose of the present work is to investigate the sense in which the Bethe–Boltzmann equation defines an ‘integrable kinetic equation’. To this end, we study a class of N dimensional systems of evolution equations that arise naturally as finite-dimensional approximations to the Bethe–Boltzmann equation. We obtain non-local Poisson brackets and Hamiltonian densities for these equations and derive an infinite family of first integrals, parameterized by N functional degrees of freedom. We find that the conserved charges arising from quantum integrability map to Casimir invariants of the hydrodynamic bracket and their group velocities map to Hamiltonian flows. Some results from the finite-dimensional setting extend to the underlying integro-differential equation, providing evidence for its integrability in the hydrodynamic sense.

Construction and simulation of a novel continuous traffic flow model

In this paper, we aim to propose a novel mathematical model for traffic flow and apply a newly developed characteristic particle method to solve the associate governing equations. As compared with the existing non-equilibrium higher-order traffic flow models, the present one is put forward to satisfy the following three conditions:1.Preserve the equilibrium state in the smooth region.2.Yield an anisotropic propagation of traffic flow information.3.Expressed with a conservation law form for traffic momentum.These conditions will ensure a more practical simulation in traffic flow physics: The current traffic will not be influenced by the condition in the behind and result in unambiguous condition across a traffic shock. Through analyses of characteristics, stability condition and steady-state solution adherent to the equation system, it is shown that the proposed model actually conform to these conditions. Furthermore, this model can be cast into its characteristic form which, incorporated with the Rankine-Hugoniot relation, is appropriate to be simulated by the characteristic particle method to obtain accurate computational results.

Time-dependent spectral renormalization method

The spectral renormalization method was introduced by Ablowitz and Musslimani (2005) as an effective way to numerically compute (time-independent) bound states for certain nonlinear boundary value problems. In this paper, we extend those ideas to the time domain and introduce a time-dependent spectral renormalization method as a numerical means to simulate linear and nonlinear evolution equations. The essence of the method is to convert the underlying evolution equation from its partial or ordinary differential form (using Duhamel’s principle) into an integral equation. The solution sought is then viewed as a fixed point in both space and time. The resulting integral equation is then numerically solved using a simple renormalized fixed-point iteration method. Convergence is achieved by introducing a time-dependent renormalization factor which is numerically computed from the physical properties of the governing evolution equation. The proposed method has the ability to incorporate physics into the simulations in the form of conservation laws or dissipation rates. This novel scheme is implemented on benchmark evolution equations: the classical nonlinear Schrödinger (NLS), integrable PT symmetric nonlocal NLS and the viscous Burgers’ equations, each of which being a prototypical example of a conservative and dissipative dynamical system. Numerical implementation and algorithm performance are also discussed.

Multilayer system of films heated from above

The paper deals with the development of convective flows in a complex system of thin liquid layers occurring under the effect of thermal load. To study this process a mathematical model which is based on the decomposition of complex problem into unitary elements (modules) with some set of rules allowing their linkage with each other was constructed. Each module represents a monotypic local model. The fluid flow and heat transfer in each of these modules are described by the Navier-Stokes and thermal conductivity equations. The boundary conditions required to solve these systems of equations are represented explicitly and written in the form of conservation laws. The numerical analysis of effect of thermal load on the characteristics of the liquid layers movement depending on the Marangoni number is carried out. It is shown that physical and thermophysical properties of liquid layers play the decisive role. It is exactly they control the convective flows in layers and determine the position and shape of interfaces. The results of model problem solutions are presented.

Time-linearized, compact methods for the inviscid GRLW equation subject to initial Gaussian conditions

• Effects of the width of initial Gaussian conditions on wave propagation are studied. • Roles of nonlinearity and local advection on wave dynamics are investigated. • Time-linearization methods for the GRLW equation are presented. • Rarefaction waves may appear behind solitary waves. • Localized dispersive oscillations may occur behind the leading wave. The objective of this paper is three-fold. First, four time-linearization methods that are second- and fourth-order accurate in time and space, respectively, are presented and used to study the dynamics of the modified and generalized regularized-long wave equations (mRLW and GRLW equations, respectively). Two of the methods use the conservation-law form of the equations and treat the wave amplitude and its second-order spatial derivative and the linear and nonlinear advection fluxes as unknowns, whereas the other two employ the non-conservation-law form of the equations and consider the wave amplitude and its first- and second-order spatial derivatives as unknowns. The methods employ three-point fourth-order accurate Padé discretizations for the first- and second-order derivatives, are second-order accurate in time, and yield linear systems of blocktridiagonal matrices. Second, the accuracy of these methods is assessed by comparing their results with those of the exact solution of the mRLW equation. It is reported that the four methods predict nearly the same values of the three invariants and have the same accuracy, and that an accurate prediction of the invariants may not correspond to small errors in space and time. Third, the dynamics of the inviscid GRLW equation is studied first qualitatively in terms of length and time scales and then numerically as a function of the linear advection speed, the exponent of the nonlinear advection flux, the dispersion coefficient and the amplitude and width of the initial bell-shaped or Gaussian conditions. It is shown that wide initial conditions result in wave steepening and breakup and the formation of solitary waves whose amplitude and speed decrease as the time for their formation increases. For narrow initial conditions, it is shown that only a single solitary wave may form. Behind this wave and depending on the parameters that characterized the inviscid GRLW equation, rarefaction or negative amplitude waves that propagate towards the upstream boundary or a train of localized oscillatory waves that do not emerge from the trailing edge of the leading solitary wave may be formed. These oscillatory waves exhibit the characteristics of, but are not dispersive shock waves and their amplitude and frequency increases as the width of the initial conditions is decreased. The results presented here do not only complement previous work by the authors, they also show that the dynamics of the inviscid GRLW equation undergoes new and interesting phenomena as the width of the initial conditions is decreased.

Numerical solution of the fully non-linear weakly dispersive serre equations for steep gradient flows

• The Serre equations are written in conservation law form and solved as a hyperbolic system. • The approach can be used with other dispersive equations. • The new model is validated using laboratory data that contain steep gradients. • First published validation of this new approach to solve the Serre equations. • Our approach is accurate, simple to implement and stable. We demonstrate a numerical approach for solving the one-dimensional non-linear weakly dispersive Serre equations. By introducing a new conserved quantity the Serre equations can be written in conservation law form, where the velocity is recovered from the conserved quantities at each time step by solving an auxiliary elliptic equation. Numerical techniques for solving equations in conservative law form can then be applied to solve the Serre equations. We demonstrate how this is achieved. The system of conservation equations are solved using the finite volume method and the associated elliptic equation for the velocity is solved using a finite difference method. This robust approach allows us to accurately solve problems with steep gradients in the flow, such as those generated by discontinuities in the initial conditions. The method is shown to be accurate, simple to implement and stable for a range of problems including flows with steep gradients and variable bathymetry.

A variationally consistent Streamline Upwind Petrov–Galerkin Smooth Particle Hydrodynamics algorithm for large strain solid dynamics

This paper presents a new Smooth Particle Hydrodynamics (SPH) computational framework for explicit fast solid dynamics. The proposed methodology explores the use of the Streamline Upwind Petrov–Galerkin (SUPG) stabilisation methodology as an alternative to the Jameson–Schmidt–Turkel (JST) stabilisation recently presented by the authors in Lee et al. (2016) in the context of a conservation law formulation of fast solid dynamics. The work introduced in this paper puts forward three advantageous features over the recent JST-SPH framework. First, the variationally consistent nature of the SUPG stabilisation allows for the introduction of a locally preserving angular momentum procedure which can be solved in a monolithic manner in conjunction with the rest of the system equations. This differs from the JST-SPH framework, where an a posteriori projection procedure was required to ensure global angular momentum preservation. Second, evaluation of expensive harmonic and bi-harmonic operators, necessary for the JST stabilisation, is circumvented in the new SUPG-SPH framework. Third, the SUPG-SPH framework is more accurate (for the same number of degrees of freedom) than its JST-SPH counterpart and its accuracy is comparable to that of the robust (but computationally more demanding) Petrov–Galerkin Finite Element Method (PG-FEM) technique explored by the authors in Lee et al. (2014), Gil et al. (2014,2016), Bonet et al. (2015), as shown in the numerical examples included. A series of numerical examples are analysed in order to benchmark and assess the robustness and effectiveness of the proposed algorithm. The resulting SUPG-SPH framework is therefore accurate, robust and computationally efficient, three key desired features that will allow the authors in forthcoming publications to explore its applicability in large scale simulations.

3D Compressible Navier-Stokes Equations Solve Optical Properties of Forced Shear Layers

The optical properties of shear layers is a topic of research with applications in high power lasers and optical imaging systems. In recent years, it has been shown that large-scale structures are intrinsic features in a plane turbulent mixing layer. While there has been some research on the effects of turbulence on laser beams, there has been no study of the optical effects of large-scale structures as they exist in shear layer development, nor how these effects may be controlled. Since the susceptibility of a free shear layer to a forcing input increases as the location of the forcing moves closer to the trailing edge of the splitter plate. Disturbances can be imposed on shear layer using very small-amplitude displacements of the splitter-plate trailing edge in a direction perpendicular to the flow direction. We are studying the optical effects of large-scale structures in forced and unforced situations using periodic motion of a small flap at the trailing edge of the splitter plate. Three-dimensional Navier-Stokes equations in conservation law form are solved directly using the WENO scheme for the simulation of a spatially-developing free shear layer.

Survey and Review

The Survey and Review paper of this issue, “Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws,” by G. A. El, M. A. Hoefer, and M. Shearer, connects two well-known areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves, and uses a raft of applied mathematics tools: phase portrait analysis, multiple scales, averaging, etc. The theory of nonlinear dispersive waves has its roots in nineteenth century studies of water waves and gained an enormous prominence after the computer discovery of the soliton by Zabusky and Kruskal in 1965. This discovery led to the theory of integrable Hamiltonian partial differential equations with ramifications in many areas of mathematics, and also in mathematical physics, nonlinear optics, etc. The evolution of the theory of hyperbolic conservation laws has followed a similar pattern; while it started with Riemann's work on gas dynamics, it only came of age in the 1950s and 1960s with the contributions of Lax, Oleinik, and others. Regardless of the smoothness of the initial data, solutions of nonlinear conservation laws typically develop discontinuities and therefore have to be understood in a weak sense. To make matters more complicated, weak solutions are often not unique, and the solution with physical relevance has to be identified with the help of an entropy condition. A way to avoid these difficulties is to regularize the differential equations. Regularization is of course not only a matter of mathematical convenience: it brings to the equation “higher order” physical effects ignored in the conservation law formulation. The dissipative regularization of the inviscid Burgers equation \( u_t+(u^2/2)_x = 0\) results in the Burgers equation \( u_t+(u^2/2)_x = \epsilon u_xx\), \(\epsilon >0\). The inviscid shock that jumps from the upstream state \( u^-\) to the downstream state \(u^+\), \(u^->u^+\), is regularized to become a viscous, smooth traveling wave solution that shares the shock speed and upstream and downstream values. In situations without dissipation, regularization may arise from dispersion, as in the Korteweg--de Vries (KdV) equation \( u_t+(u^2/2)_x = \mu u_xxx\), and in that case the regularization of shock solutions leads to so-called dispersive shock waves, entities with a structure much richer than that of the dissipative traveling wave. The paper focuses on the modified KdV Burgers equation \( u_t+(u^3)_x = \epsilon u_xx+\mu u_xxx\), a universal asymptotic model where the corresponding conservation law \( u_t+(u^3)_x = 0\) has a nonconvex flow \( u^3\). As the paper shows, many remarkable phenomena appear due to the competition among nonconvexity, dissipation, and dispersion; these include undercompressive shock waves and double-wave complexes such as shock-rarefactions.

Solution of an inverse scattering problem for the acoustic wave equation in three-dimensional media

A three-dimensional inverse scattering problem for the acoustic wave equation is studied. The task is to determine the density and acoustic impedance of a medium. A necessary and sufficient condition for the unique solvability of this problem is established in the form of an energy conservation law. The interpretation of the solution to the inverse problem and the construction of medium images are discussed.

Re-appraisal and extension of the Gratton-Vargas two-dimensional analytical snowplow model of plasma focus. III. Scaling theory for high pressure operation and its implications

Recent work on the revised Gratton-Vargas model (Auluck, Phys. Plasmas 20, 112501 (2013); 22, 112509 (2015) and references therein) has demonstrated that there are some aspects of Dense Plasma Focus (DPF), which are not sensitive to details of plasma dynamics and are well captured in an oversimplified model assumption, which contains very little plasma physics. A hyperbolic conservation law formulation of DPF physics reveals the existence of a velocity threshold related to specific energy of dissociation and ionization, above which, the work done during shock propagation is adequate to ensure dissociation and ionization of the gas being ingested. These developments are utilized to formulate an algorithmic definition of DPF optimization that is valid in a wide range of applications, not limited to neutron emission. This involves determination of a set of DPF parameters, without performing iterative model calculations, that lead to transfer of all the energy from the capacitor bank to the plasma at the time of current derivative singularity and conversion of a preset fraction of this energy into magnetic energy, while ensuring that electromagnetic work done during propagation of the plasma remains adequate for dissociation and ionization of neutral gas being ingested. Such a universal optimization criterion is expected to facilitate progress in new areas of DPF research that include production of short lived radioisotopes of possible use in medical diagnostics, generation of fusion energy from aneutronic fuels, and applications in nanotechnology, radiation biology, and materials science. These phenomena are expected to be optimized for fill gases of different kinds and in different ranges of mass density compared to the devices constructed for neutron production using empirical thumb rules. A universal scaling theory of DPF design optimization is proposed and illustrated for designing devices working at one or two orders higher pressure of deuterium than the current practice of designs optimized at pressures less than 10 mbar of deuterium. These examples show that the upper limit for operating pressure is of technological (and not physical) origin.

Local isometric immersions of pseudospherical surfaces described by evolution equations in conservation law form

For the pseudospherical surfaces described by a class of second order evolution equations, of the form zt=A(x,t,z)z2+B(x,t,z,z1), we consider the problem of local isometric immersion into the 3-dimensional Euclidean space E3 with a second fundamental form depending on finite-order jets of solutions z of the considered equations. We also provide an extension of our analysis to the case of k-th order evolution equations in conservation law form. Examples of equations admitting such local isometric immersions, are provided by equations like Burgers, Murray, Svinolupov–Sokolov, Kuramoto–Sivashinsky, Sawada–Kotera, Kaup–Kupershmidt, as well as hierarchies of evolution equations in conservation law form like Burgers, mKdV and KdV.

Dual number coefficient octonion algebra, field equations and conservation laws

Starting with octonion algebra, we develop the dual number coefficient octonion (DNCO) algebra having sixteen components. DNCO forms of generalized potential, field and current equations are discussed in consistent manner. We have made an attempt to write the DNCO form of generalized Dirac–Maxwell’s equations in presence of electric and magnetic charges (dyons). Accordingly, we demonstrate the work-energy theorem of classical mechanics reproducing the continuity equation for dyons in terms of DNCO algebra. Further, we discuss the DNCO form of linear momentum conservation law for dyons.

A solution of the conservation law form of the Serre equations

The nonlinear and weakly dispersive Serre equations contain higher-order dispersive terms. These include mixed spatial and temporal derivative flux terms which are difficult to handle numerically. These terms can be replaced by an alternative combination of equivalent temporal and spatial terms, so that the Serre equations can be written in conservation law form. The water depth and new conserved quantities are evolved using a second-order finite-volume scheme. The remaining primitive variable, the depth-averaged horizontal velocity, is obtained by solving a second-order elliptic equation using simple finite differences. Using an analytical solution and simulating the dam-break problem, the proposed scheme is shown to be accurate, simple to implement and stable for a range of problems, including flows with steep gradients. It is only slightly more computationally expensive than solving the shallow water wave equations. doi:10.1017/S144618111600002X

Conservation of energy–momentum tensor in fermionic superfluid phase: Effects of U(1) broken symmetry

Respecting the conservation laws of momentum and energy in a many body theory is very important for understanding the transport phenomena. The previous conserving approximation requires that the self-energy of a single particle could be written as a functional derivative of a full dressed Green's function. This condition can not be satisfied in the G0G t-matrix or pair fluctuation theory which emphasizes the fermion pairing with a stronger than the Bardeen–Cooper–Schrieffer (BCS) attraction. In the previous work [1], we have shown that when the temperature is above the superfluid transition temperature Tc, the G0G t-matrix theory can be put into a form that satisfies the stress tensor Ward identity (WI) or local form of conservation laws by introducing a new type of vertex correction. In this paper, we will extend the above conservation approximation to the superfluid phase in the BCS mean field level. To establish the stress tensor WI, we have to include the fluctuation of the order parameter or the contribution from the Goldstone mode. The result will be useful for understanding the transport properties such as the behavior of the viscosity of Fermionic gases in the superfluid phases.

A SOLUTION OF THE CONSERVATION LAW FORM OF THE SERRE EQUATIONS

The nonlinear and weakly dispersive Serre equations contain higher-order dispersive terms. These include mixed spatial and temporal derivative flux terms which are difficult to handle numerically. These terms can be replaced by an alternative combination of equivalent temporal and spatial terms, so that the Serre equations can be written in conservation law form. The water depth and new conserved quantities are evolved using a second-order finite-volume scheme. The remaining primitive variable, the depth-averaged horizontal velocity, is obtained by solving a second-order elliptic equation using simple finite differences. Using an analytical solution and simulating the dam-break problem, the proposed scheme is shown to be accurate, simple to implement and stable for a range of problems, including flows with steep gradients. It is only slightly more computationally expensive than solving the shallow water wave equations.